A moving dot P departs from O at an initial speed of 6 m/s in the east direction and accelerates at 2 m/s^2 in the east direction. 2 seconds after P's departure, dot Q departs from O to chase down P at a constant speed of k m/s in the east direction. What is the minimum speed, k, required for Q to eventually catch up to P?
P: s = 6t + t^2
Q: s = k(t-2)
we need to solve for when the distances are equal:
k(t-2) = 6t + t^2
t^2 + (6-k)t + 2k = 0
t = [(k-6)±√((6-k)^2-8k)]/2
for this to have real values,
(6-k)^2-8k >= 0
k^2 - 20k + 36 >= 0
k >= 18
so, if k=18, Q overtakes P at t=6
check:
P(6) = 72
Q(6) = 18*4 = 72
For k>18, Q overtakes P sooner